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On the Existence and Uniqueness of Static, Spherically Symmetric Stellar Models in General Relativity

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On the Existence and Uniqueness of Static, Spherically Symmetric Stellar Models in General Relativity University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Masters Theses Graduate School 8-2015 On the Existence and Uniqueness of Static, Spherically Symmetric Stellar Models in General Relativity Josh Michael Lipsmeyer University of Tennessee - Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_gradthes Part of the Cosmology, Relativity, and Gravity Commons, and the Geometry and Topology Commons Recommended Citation Lipsmeyer, Josh Michael, "On the Existence and Uniqueness of Static, Spherically Symmetric Stellar Models in General Relativity. " Master's Thesis, University of Tennessee, 2015. https://trace.tennessee.edu/utk_gradthes/3490 This Thesis is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Masters Theses by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a thesis written by Josh Michael Lipsmeyer entitled "On the Existence and Uniqueness of Static, Spherically Symmetric Stellar Models in General Relativity." I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Master of Science, with a major in Mathematics. Alex Freire, Major Professor We have read this thesis and recommend its acceptance: Tadele Mengesha, Mike Frazier Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) On the Existence and Uniqueness of Static, Spherically Symmetric Stellar Models in General Relativity A Thesis Presented for the Master of Science Degree The University of Tennessee, Knoxville Josh Michael Lipsmeyer August 2015 c by Josh Michael Lipsmeyer, 2015 All Rights Reserved. ii I would like to dedicate this work to my family: Mom, Dad, Joey, Brandi, Tim, and Shannon Your love, support, and encouragement has never waned for one second and for that I am truly thankful. iii Acknowledgments I would like to thank my advisor, Dr. Alex Freire, for his insights, guidance, and encouragement. I would like to also thank my committee members Dr. Mike Frazier and Dr. Tadele Mengesha for their contribution to my development in mathematics. I would also like to thank the geometry group at the University of Tennessee for always making themselves available to talk through problems and offer their own valuable insights. Finally, I would like to thank Dr. Tom McMillan from the University of Arkansas at Little Rock for the pivotal role he played in laying the foundation of my career in mathematics, his constant encouragement, and friendship. iv \Everybody can be great. Because anybody can serve. You don't have to have a college degree to serve. You don't have to make your subject and your verb agree to serve. You don't have to know about Plato and Aristotle to serve. You don't have to know Einstein's theory of relativity to serve. You don't have to know the second theory of thermodynamics to serve. You only need a heart full of grace. A soul generated by love." - Martin Luther King, Jr. \We all die. The goal isn't to live forever, the goal is to create something that will." - Chuck Palahniuk v Abstract The "Fluid Ball Conjecture" states that a static stellar model in General Relativity is spherically symmetric. This conjecture has been the motivation of much work since first studied by Avez in 1964. There have been many partial results( ul-Alam, Lindblom, Beig and Simon, etc) which rely heavily on arguments using the Positive Mass Theorem and the equivalence of conformal flatness and spherical symmetry. The purpose of this thesis is to outline the general problem, analyze and compare the key differences in several of the partial results, and give existence and uniqueness proofs for a particular class of equations of state which represents the most recent progress towards a fully generalized solution. vi Table of Contents 1 Introduction 1 2 Existence and Uniqueness of Static, Spherically Symmetric Solu- tions 5 2.1 Derivation of the System of Equations . 6 2.2 The Singular Point . 17 2.3 Extending the Solution Uniquely . 19 3 Uniqueness of the Static Stellar Model 23 3.1 Statement of the theorem. 29 3.2 The \reference system" of O.D.E. 31 3.3 Spinor approach to the Positive Mass Theorem. 32 3.4 Strategy of proof of the Main Theorem. 34 3.5 Auxiliary system of differential equations. 38 3.6 The critical set and the oscillation set. 41 3.7 The conformal factor on the regular set. 46 3.7.1 Accumulation of discontinuities. 49 3.8 Convergence of a sequence of conformal factors. 51 3.9 Conclusion of the proof. 54 4 Physical Constraints on Stellar Models 64 4.1 Buchdahl's Inequality . 65 vii 4.2 Constraints on the Adiabatic Index . 70 4.3 Necessary and Sufficient Conditions for a Finite Radius . 76 Bibliography 81 Appendix 86 A Spin Geometry 87 A.1 Dirac Spinors . 87 A.2 Spin Manifold and the Differential Structure of the Dirac Bundle . 92 A.3 Dirac Operator and the Schr¨odinger-Lichnerowicz formula . 97 A.4 Witten's expression for the mass . 102 Vita 108 viii Chapter 1 Introduction The \Fluid Ball Conjecture" seems to have first been addressed by Avez in 1964.[23] The conjecture is concerned with equilibrium configurations for static stellar models. As in the Newtonian case [21], the belief is that a static stellar model equilibrium configuration always attains spherical symmetry. Partial results to the conjecture were attained in the 1970's and 1980's. Kunzle [20], Kunzle and Savage [15] made contributions to the conjecture but used very restrictive equations of state. ul-Alam [24],[25] brought in the use of the Posititve Mass Theorem [28]. ul-Alam's work in the 1980's unfortunately relied on unphysical equations of state to complete the proof. Lindblom [22] succeeded in proving spherical symmetry in the case of uniform density stars using ul-Alam's work and the use of Robinson-type identities that had previously been used in proving uniqueness of static black holes.[27],[26]. A drawback to the work of ul-Alam and Lindblom in the 1980's was that their method of proof required the existence of a \reference spherical stellar model" with the same mass and surface potential Vs as their static stellar model. To complete the proof using their method they assumed the existence of the \reference model", without proof. In 1994 joint work of ul-Alam and Lindblom [29] proved the existence of the "reference spherical stellar model" under certain restrictions on the equation of state. This represented the most complete work up to that point. Given a static stellar model 1 and assuming an equation of state satisfying certain properties, the \reference stellar model" existed and the procedure developed in the 1980's utilizing the Positive Mass Theorem showed that the static stellar model must in fact be spherical. In 2007 the most recent result pertaining to the \Fluid Ball Conjecture" by ul-Alam used a spinor norm weighted scalar curvature integral. It was this integral that had been used by Witten to prove the Positive Energy Theorem in n-dimensions[30]. The method of proving spherical symmetry using the Positive Mass Theorem has been the standard method of proof since it was first utilized by ul-Alam. The procedure is to start with a static stellar model with a certain given equation of state. The goal is to find a conformal factor so that the mass of the conformal metric is zero and the conformal scalar curvature is non-negative. The Positive Mass Theorem then implies that the conformal metric must in fact be flat. In the conformally flat case , Avez [23] showed that the original geometry had to be spherical. The difficulty in this method is showing the non-negativity of the conformal scalar curvature. The conformal factor is modeled after the conformal factor for the \reference spherical model". This was the source of difficulty in ul-Alam and Lindblom's work in the 1980's. In order to show existence of the \reference stellar model" and the non- negativity of the conformal scalar curvature certain restrictions were placed on the equation of state. All of the modifications to this method revolved around restrictions on the equation of state. Point-wise non-negativity of the conformal scalar curvature is a strict requirement. In an effort to relax this condition, the use of a spinor norm weighted scalar curvature integral was utilized by ul-Alam in 2007. This allows the point-wise non-negativity to be relaxed as long as the overall negative contribution to the integral of the conformal scalar curvature is small. The scalar curvature integral is precisely the integral used by Witten in his proof of the Positive Energy Theorem. In ul-Alam's work a conformal 2 factor is defined as a limit of conformal factors. In the limit, the scalar curvature integral with the scalar curvature in the original metric is shown to go to zero. The scalar curvature integral equaling zero implies the existence of a global convariantly constant spinor field. It is known that spinors are a type of \square root" of a vector so the global covariantly constant spinor field allows us to define a global covariantly constant frame field. This implies that the space is flat. Classical arguments [23] then imply that since the conformal geometry is flat, then the original geometry must be spherically symmetric.
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